In
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matric ...
, a multilinear map is a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-orie ...
of several variables that is
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
separately in each variable. More precisely, a multilinear map is a function
:
where
and
are
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
s (or
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Mo ...
s over a
commutative ring), with the following property: for each
, if all of the variables but
are held constant, then
is a
linear function
In mathematics, the term linear function refers to two distinct but related notions:
* In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For di ...
of
.
A multilinear map of one variable is a
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
, and of two variables is a
bilinear map
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example.
Definition
Vector spaces
Let V, ...
. More generally, a multilinear map of ''k'' variables is called a ''k''-linear map. If the
codomain
In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either ...
of a multilinear map is the
field of scalars
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
, it is called a
multilinear form
In abstract algebra and multilinear algebra, a multilinear form on a vector space V over a field K is a map
:f\colon V^k \to K
that is separately ''K''-linear in each of its ''k'' arguments. More generally, one can define multilinear forms on ...
. Multilinear maps and multilinear forms are fundamental objects of study in
multilinear algebra
Multilinear algebra is a subfield of mathematics that extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of '' ...
.
If all variables belong to the same space, one can consider
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
,
antisymmetric and
alternating
Alternating may refer to:
Mathematics
* Alternating algebra, an algebra in which odd-grade elements square to zero
* Alternating form, a function formula in algebra
* Alternating group, the group of even permutations of a finite set
* Alter ...
''k''-linear maps. The latter coincide if the underlying
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
(or
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
) has a
characteristic different from two, else the former two coincide.
Examples
* Any
bilinear map
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example.
Definition
Vector spaces
Let V, ...
is a multilinear map. For example, any
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
on a vector space is a multilinear map, as is the
cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and i ...
of vectors in
.
* The
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if ...
of a matrix is an
alternating
Alternating may refer to:
Mathematics
* Alternating algebra, an algebra in which odd-grade elements square to zero
* Alternating form, a function formula in algebra
* Alternating group, the group of even permutations of a finite set
* Alter ...
multilinear function of the columns (or rows) of a
square matrix
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are ofte ...
.
* If
is a
''Ck'' function, then the
th derivative of
at each point
in its domain can be viewed as a
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
-linear function
.
Coordinate representation
Let
:
be a multilinear map between finite-dimensional vector spaces, where
has dimension
, and
has dimension
. If we choose a
basis
Basis may refer to:
Finance and accounting
*Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting o ...
for each
and a basis
for
(using bold for vectors), then we can define a collection of scalars
by
:
Then the scalars
completely determine the multilinear function
. In particular, if
:
for
, then
:
Example
Let's take a trilinear function
:
where , and .
A basis for each is
Let
:
where
. In other words, the constant
is a function value at one of the eight possible triples of basis vectors (since there are two choices for each of the three
), namely:
:
Each vector
can be expressed as a linear combination of the basis vectors
:
The function value at an arbitrary collection of three vectors
can be expressed as
:
Or, in expanded form as
:
Relation to tensor products
There is a natural one-to-one correspondence between multilinear maps
:
and linear maps
:
where
denotes the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...
of
. The relation between the functions
and
is given by the formula
:
Multilinear functions on ''n''×''n'' matrices
One can consider multilinear functions, on an matrix over a
commutative ring with identity, as a function of the rows (or equivalently the columns) of the matrix. Let be such a matrix and , be the rows of . Then the multilinear function can be written as
:
satisfying
:
If we let
represent the th row of the identity matrix, we can express each row as the sum
:
Using the multilinearity of we rewrite as
:
Continuing this substitution for each we get, for ,
:
Therefore, is uniquely determined by how operates on
.
Example
In the case of 2×2 matrices we get
:
Where